In the song "You're So Vain" by Carly Simon, supposedly written about Warren Beatty, there's a horse that Warren Beatty backs in Saratoga which "naturally won." What was the name of the horse?

There is a whole page on Carly Simon's website about this song. If she won't even divulge the name of the man that the song is about, what makes you think we'll ever get the name of the horse? Or maybe 'Naturally' is the name of the horse!
[carlysimon.com/vain/vain.htm] Link no longer exists

There has been much debate over who exactly the song is about. Popular guesses on the subject include Mick Jagger (who sang backing vocals on the song), Cat Stevens, Warren Beatty, Kris Kristofferson, unfaithful fiancé William Donaldson and Simon's ex-husband, James Taylor. Despite these guesses and many interviews asking who it was, to this day Carly Simon has never publicly acknowledged in full who the song is about.

Dick Ebersol, president of NBC Sports and a friend of Simon, was the highest bidder for a Martha's Vineyard Possible Dreams charity auction offering in which the prize was the revelation of the person whom "You're So Vain" was about. A condition of that prize is that Ebersol would not reveal to anyone the actual subject. Later, Ebersol revealed that he was allowed by Simon to divulge a clue about the person's name: "Carly told me that I could offer up to the entire world, a clue as to what she'll tell me when we have this night in about two weeks. And the clue is: the letter 'E' is in the person's name." Dick Ebersol on NBC's Today Show - August 5, 2003

https://en.wikipedia.org/wiki/You're_So_Vain

If the line is punctuated in this way, the horse's name could be "Naturally."

...and your horse, Naturally, won.

I have the old Carly Simon album, The Best of Carly Simon, from 1975. It includes all the lyrics. McGruff is absolutely right; however, there are no commas, so there is no mention of the horse's name in my interpretation.

In a 2015 interview, Carly Simon confirmed that the second verse of her most famous song is about Warren Beatty, though the remaining verses (including the one about the horse) are about two other men whom she has so far declined to name.

http://www.latimes.com/entertainment/gossip/la-et-mg-carly-simon-youre-so-vain-about-warren-beatty-partially-20151118-story.html